If $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$

Question

I've been thinking about the following claim:

Let $A$ be a set and $|A|$ his cardinality. For every cardinal $\lambda$ with $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$.

This is true? Is there some relation between this assertion and CH or GCH?

Answer 1

If $\lambda<|A|$, there is an injection $h:\lambda\to A$, and the set $h[\lambda]=\operatorname{ran}h$ is a subset of $A$ with cardinality $\lambda$. This has nothing to do with $\mathsf{(G)CH}$.